{"id":563,"date":"2025-10-09T15:22:22","date_gmt":"2025-10-09T15:22:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=563"},"modified":"2025-10-23T12:09:16","modified_gmt":"2025-10-23T12:09:16","slug":"motion-in-space-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/motion-in-space-learn-it-1\/","title":{"raw":"Motion in Space: Learn It 1","rendered":"Motion in Space: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Describe how particles move in space using velocity and acceleration vectors<\/li>\r\n \t<li>Explain the tangential and normal components of acceleration<\/li>\r\n \t<li>Use vector-valued functions to describe and analyze planetary motion<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Motion Vectors in the Plane and in Space<\/h2>\r\n<p class=\"whitespace-normal break-words\">You now know how to describe curves in the plane and in space, and how to determine their properties like arc length and curvature. These tools come together to describe motion along curves\u2014the main goal of this chapter.<\/p>\r\n<p class=\"whitespace-normal break-words\">Our starting point is using vector-valued functions to represent the position of an object as a function of time. This approach works for both plane curves and space curves. For example, planetary orbits are elliptical curves that lie in a plane, while a particle traveling along a helix moves through three-dimensional space.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>position, velocity, acceleration, and speed<\/h3>\r\n<p class=\"whitespace-normal break-words\">Let [latex]{\\bf{r}}(t)[\/latex] be a twice-differentiable vector-valued function of the parameter [latex]t[\/latex] that represents the position of an object as a function of time.<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Velocity vector:<\/strong> [latex]{\\bf{v}}(t)={\\bf{r}}'(t)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Acceleration vector:<\/strong> [latex]{\\bf{a}}(t)={\\bf{v}}'(t)={\\bf{r}}''(t)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Speed:<\/strong> [latex]v(t)=\\left\\Vert{\\bf{v}}(t)\\right\\Vert=\\left\\Vert{\\bf{r}}'(t)\\right\\Vert=\\frac{ds}{dt}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Since [latex]{\\bf{r}}(t)[\/latex] can describe motion in either two or three dimensions, these vector-valued functions can have either two or three components. In two dimensions, we write [latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}[\/latex], and in three dimensions [latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}+z(t){\\bf{k}}[\/latex]. The table below shows how velocity, acceleration, and speed are expressed in each case.<\/p>\r\n\r\n<table class=\"text-center w-full\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\"><strong>Quantity<\/strong><\/th>\r\n<th style=\"text-align: center;\"><strong>Two Dimensions<\/strong><\/th>\r\n<th style=\"text-align: center;\"><strong>Three Dimensions<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">Position<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}+z(t){\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Velocity<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{v}}(t)=x'(t){\\bf{i}}+y'(t){\\bf{j}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{v}}(t)=x'(t){\\bf{i}}+y'(t){\\bf{j}}+z'(t){\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Acceleration<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{a}}(t)=x''(t){\\bf{i}}+y''(t){\\bf{j}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]{\\bf{a}}(t)=x''(t){\\bf{i}}+y''(t){\\bf{j}}+z''(t){\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Speed<\/td>\r\n<td style=\"text-align: center;\">[latex]v(t)=\\sqrt{(x'(t))^{2}+(y'(t))^{2}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]v(t)=\\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\" aria-label=\"Example\">A particle moves in a parabolic path defined by the vector-valued function [latex]{\\bf{r}}\\,(t)=t^{2}\\,{\\bf{i}}+\\sqrt{5-t^{2}}\\,{\\bf{j}}[\/latex], where [latex]t[\/latex] measures time in seconds.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Find the velocity, acceleration, and speed as functions of time.<\/li>\r\n \t<li>Sketch the curve along with the velocity vector at time [latex]t=1[\/latex].<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1167793461764\" class=\"exercise\">[reveal-answer q=\"fs-id1167795055169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167795055169\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>We use our definition:\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {{\\bf{v}}\\,(t)} &amp; =\\hfill &amp; {{\\bf{r}}'\\,(t)=2t\\,{\\bf{i}}-\\frac{t}{\\sqrt{5-t^{2}}}\\,{\\bf{j}}} \\hfill \\\\ \\hfill {{\\bf{a}}\\,(t)} &amp; =\\hfill &amp; {{\\bf{v}}'\\,(t)=2\\,{\\bf{i}}-5(5-t^{2})^{-3\/2}\\,{\\bf{j}}} \\hfill \\\\ \\hfill {v\\,(t)} &amp; =\\hfill &amp; {\\left\\Vert{\\bf{r}}'\\,(t)\\right\\Vert} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\sqrt{(2t)^{2}+\\Big(-\\frac{t}{\\sqrt{5-t^{2}}}\\Big)^{2}}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\sqrt{4t^{2}+\\frac{t^{2}}{5-t^{2}}}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\sqrt{\\frac{21t^{2}-4t^{4}}{5-t^{2}}}.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;<\/li>\r\n \t<li>The graph of [latex]{\\bf{r}}\\,(t)=t^{2}\\,{\\bf{I}}+\\sqrt{5-t^{2}}\\,{\\bf{j}}[\/latex] is a portion of a parabola (Figure 1). The velocity vector at [latex]t=1[\/latex] is\r\n<div style=\"text-align: center;\">[latex]{\\bf{v}}\\,(1)={\\bf{r}}'\\,(1)=2\\,(1)\\,{\\bf{i}}-\\frac{1}{\\sqrt{5-(1)^{2}}}\\,{\\bf{j}}=2\\,{\\bf{i}}-\\frac{1}{2}\\,{\\bf{j}}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nand the acceleration vector at [latex]t=1[\/latex] is\r\n<div style=\"text-align: center;\">[latex]{\\bf{a}}\\,(1)={\\bf{v}}'\\,(t)=2\\,{\\bf{i}}-5(5-(1)^{2})^{-3\/2}\\,{\\bf{j}}=2\\,{\\bf{i}}-\\frac{5}{8}\\,{\\bf{j}}.[\/latex]<\/div>\r\n&nbsp;\r\n\r\nNotice that the velocity vector is tangent to the path, as is always the case.\r\n<div>[caption id=\"attachment_923\" align=\"aligncenter\" width=\"342\"]<img class=\"size-full wp-image-923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162519\/3-4-1.jpeg\" alt=\"This figure is the graph of a curve in the first quadrant. The curve begins on the y axis slightly above y=2 and decreases to the x-axis at x=5. On the curve is a tangent vector labeled \u201cv(1)\u201d and is pointing towards the x-axis.\" width=\"342\" height=\"346\" \/> Figure 1.\u00a0This graph depicts the velocity vector at time [latex]t=1[\/latex] for a particle moving in a parabolic path.[\/caption]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Understanding Velocity and Acceleration Vectors<\/h3>\r\nImagine driving along a curvy road. If you don't turn the steering wheel, you continue in a straight line and run off the road. The speed and direction at which you leave the road give a vector representing your velocity (Figure 2).\r\n\r\n[caption id=\"attachment_925\" align=\"aligncenter\" width=\"731\"]<img class=\"size-full wp-image-925\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162609\/3-4-2.jpeg\" alt=\"This figure represents a curving road. On the road is a car. At the car there are two vectors. The first vector is tangent to the back of the car. The second vector comes out of the front of the car in the direction the car is heading. Both of the vectors are labeled \u201cvelocity vectors\u201d.\" width=\"731\" height=\"306\" \/> Figure 2.\u00a0At each point along a road traveled by a car, the velocity vector of the car is tangent to the path traveled by the car.[\/caption]\r\n<p class=\"whitespace-normal break-words\">The fact that you must turn the steering wheel to stay on the road shows that your velocity is always changing\u2014even if your speed stays constant\u2014because your direction constantly changes. As you turn right, your acceleration vector points right. As you turn left, your acceleration vector points left. Your velocity and acceleration vectors are constantly changing, regardless of whether your speed varies (Figure 3).<\/p>\r\n\r\n\r\n[caption id=\"attachment_926\" align=\"aligncenter\" width=\"561\"]<img class=\"size-full wp-image-926\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162652\/3-4-3.jpeg\" alt=\"This figure is a car. The path the car is traveling is an increasing curve represented by a dotted line. The center of the car is labeled \u201ctsub0\u201d on the curve. From this point there are two vectors that are orthogonal to each other. The first vector is asubt and the second vector is asubn. In between these two vectors is a vector labeled \u201ca\u201d. It has angle theta between vector a and asubt.\" width=\"561\" height=\"392\" \/> Figure 3.\u00a0The dashed line represents the trajectory of an object (a car, for example). The acceleration vector points toward the inside of the turn at all times.[\/caption]\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\"><strong>Key Insight About Velocity and Acceleration<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">The <strong>velocity vector<\/strong> is always tangent to the path of motion<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The <strong>acceleration vector<\/strong> points toward the direction the object is turning<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Speed can remain constant even while velocity changes, because velocity depends on both speed and direction<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Describe how particles move in space using velocity and acceleration vectors<\/li>\n<li>Explain the tangential and normal components of acceleration<\/li>\n<li>Use vector-valued functions to describe and analyze planetary motion<\/li>\n<\/ul>\n<\/section>\n<h2>Motion Vectors in the Plane and in Space<\/h2>\n<p class=\"whitespace-normal break-words\">You now know how to describe curves in the plane and in space, and how to determine their properties like arc length and curvature. These tools come together to describe motion along curves\u2014the main goal of this chapter.<\/p>\n<p class=\"whitespace-normal break-words\">Our starting point is using vector-valued functions to represent the position of an object as a function of time. This approach works for both plane curves and space curves. For example, planetary orbits are elliptical curves that lie in a plane, while a particle traveling along a helix moves through three-dimensional space.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>position, velocity, acceleration, and speed<\/h3>\n<p class=\"whitespace-normal break-words\">Let [latex]{\\bf{r}}(t)[\/latex] be a twice-differentiable vector-valued function of the parameter [latex]t[\/latex] that represents the position of an object as a function of time.<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Velocity vector:<\/strong> [latex]{\\bf{v}}(t)={\\bf{r}}'(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Acceleration vector:<\/strong> [latex]{\\bf{a}}(t)={\\bf{v}}'(t)={\\bf{r}}''(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Speed:<\/strong> [latex]v(t)=\\left\\Vert{\\bf{v}}(t)\\right\\Vert=\\left\\Vert{\\bf{r}}'(t)\\right\\Vert=\\frac{ds}{dt}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\">Since [latex]{\\bf{r}}(t)[\/latex] can describe motion in either two or three dimensions, these vector-valued functions can have either two or three components. In two dimensions, we write [latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}[\/latex], and in three dimensions [latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}+z(t){\\bf{k}}[\/latex]. The table below shows how velocity, acceleration, and speed are expressed in each case.<\/p>\n<table class=\"text-center w-full\">\n<thead>\n<tr>\n<th style=\"text-align: center;\"><strong>Quantity<\/strong><\/th>\n<th style=\"text-align: center;\"><strong>Two Dimensions<\/strong><\/th>\n<th style=\"text-align: center;\"><strong>Three Dimensions<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Position<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{r}}(t)=x(t){\\bf{i}}+y(t){\\bf{j}}+z(t){\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Velocity<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{v}}(t)=x'(t){\\bf{i}}+y'(t){\\bf{j}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{v}}(t)=x'(t){\\bf{i}}+y'(t){\\bf{j}}+z'(t){\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Acceleration<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{a}}(t)=x''(t){\\bf{i}}+y''(t){\\bf{j}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{\\bf{a}}(t)=x''(t){\\bf{i}}+y''(t){\\bf{j}}+z''(t){\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Speed<\/td>\n<td style=\"text-align: center;\">[latex]v(t)=\\sqrt{(x'(t))^{2}+(y'(t))^{2}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]v(t)=\\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\" aria-label=\"Example\">A particle moves in a parabolic path defined by the vector-valued function [latex]{\\bf{r}}\\,(t)=t^{2}\\,{\\bf{i}}+\\sqrt{5-t^{2}}\\,{\\bf{j}}[\/latex], where [latex]t[\/latex] measures time in seconds.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Find the velocity, acceleration, and speed as functions of time.<\/li>\n<li>Sketch the curve along with the velocity vector at time [latex]t=1[\/latex].<\/li>\n<\/ol>\n<div id=\"fs-id1167793461764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167795055169\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167795055169\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>We use our definition:\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {{\\bf{v}}\\,(t)} & =\\hfill & {{\\bf{r}}'\\,(t)=2t\\,{\\bf{i}}-\\frac{t}{\\sqrt{5-t^{2}}}\\,{\\bf{j}}} \\hfill \\\\ \\hfill {{\\bf{a}}\\,(t)} & =\\hfill & {{\\bf{v}}'\\,(t)=2\\,{\\bf{i}}-5(5-t^{2})^{-3\/2}\\,{\\bf{j}}} \\hfill \\\\ \\hfill {v\\,(t)} & =\\hfill & {\\left\\Vert{\\bf{r}}'\\,(t)\\right\\Vert} \\hfill \\\\ \\hfill & =\\hfill & {\\sqrt{(2t)^{2}+\\Big(-\\frac{t}{\\sqrt{5-t^{2}}}\\Big)^{2}}}\\hfill \\\\ \\hfill & =\\hfill & {\\sqrt{4t^{2}+\\frac{t^{2}}{5-t^{2}}}}\\hfill \\\\ \\hfill & =\\hfill & {\\sqrt{\\frac{21t^{2}-4t^{4}}{5-t^{2}}}.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/li>\n<li>The graph of [latex]{\\bf{r}}\\,(t)=t^{2}\\,{\\bf{I}}+\\sqrt{5-t^{2}}\\,{\\bf{j}}[\/latex] is a portion of a parabola (Figure 1). The velocity vector at [latex]t=1[\/latex] is\n<div style=\"text-align: center;\">[latex]{\\bf{v}}\\,(1)={\\bf{r}}'\\,(1)=2\\,(1)\\,{\\bf{i}}-\\frac{1}{\\sqrt{5-(1)^{2}}}\\,{\\bf{j}}=2\\,{\\bf{i}}-\\frac{1}{2}\\,{\\bf{j}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>and the acceleration vector at [latex]t=1[\/latex] is<\/p>\n<div style=\"text-align: center;\">[latex]{\\bf{a}}\\,(1)={\\bf{v}}'\\,(t)=2\\,{\\bf{i}}-5(5-(1)^{2})^{-3\/2}\\,{\\bf{j}}=2\\,{\\bf{i}}-\\frac{5}{8}\\,{\\bf{j}}.[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that the velocity vector is tangent to the path, as is always the case.<\/p>\n<div>\n<figure id=\"attachment_923\" aria-describedby=\"caption-attachment-923\" style=\"width: 342px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162519\/3-4-1.jpeg\" alt=\"This figure is the graph of a curve in the first quadrant. The curve begins on the y axis slightly above y=2 and decreases to the x-axis at x=5. On the curve is a tangent vector labeled \u201cv(1)\u201d and is pointing towards the x-axis.\" width=\"342\" height=\"346\" \/><figcaption id=\"caption-attachment-923\" class=\"wp-caption-text\">Figure 1.\u00a0This graph depicts the velocity vector at time [latex]t=1[\/latex] for a particle moving in a parabolic path.<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Understanding Velocity and Acceleration Vectors<\/h3>\n<p>Imagine driving along a curvy road. If you don&#8217;t turn the steering wheel, you continue in a straight line and run off the road. The speed and direction at which you leave the road give a vector representing your velocity (Figure 2).<\/p>\n<figure id=\"attachment_925\" aria-describedby=\"caption-attachment-925\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-925\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162609\/3-4-2.jpeg\" alt=\"This figure represents a curving road. On the road is a car. At the car there are two vectors. The first vector is tangent to the back of the car. The second vector comes out of the front of the car in the direction the car is heading. Both of the vectors are labeled \u201cvelocity vectors\u201d.\" width=\"731\" height=\"306\" \/><figcaption id=\"caption-attachment-925\" class=\"wp-caption-text\">Figure 2.\u00a0At each point along a road traveled by a car, the velocity vector of the car is tangent to the path traveled by the car.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">The fact that you must turn the steering wheel to stay on the road shows that your velocity is always changing\u2014even if your speed stays constant\u2014because your direction constantly changes. As you turn right, your acceleration vector points right. As you turn left, your acceleration vector points left. Your velocity and acceleration vectors are constantly changing, regardless of whether your speed varies (Figure 3).<\/p>\n<figure id=\"attachment_926\" aria-describedby=\"caption-attachment-926\" style=\"width: 561px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-926\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21162652\/3-4-3.jpeg\" alt=\"This figure is a car. The path the car is traveling is an increasing curve represented by a dotted line. The center of the car is labeled \u201ctsub0\u201d on the curve. From this point there are two vectors that are orthogonal to each other. The first vector is asubt and the second vector is asubn. In between these two vectors is a vector labeled \u201ca\u201d. It has angle theta between vector a and asubt.\" width=\"561\" height=\"392\" \/><figcaption id=\"caption-attachment-926\" class=\"wp-caption-text\">Figure 3.\u00a0The dashed line represents the trajectory of an object (a car, for example). The acceleration vector points toward the inside of the turn at all times.<\/figcaption><\/figure>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\"><strong>Key Insight About Velocity and Acceleration<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">The <strong>velocity vector<\/strong> is always tangent to the path of motion<\/li>\n<li class=\"whitespace-normal break-words\">The <strong>acceleration vector<\/strong> points toward the direction the object is turning<\/li>\n<li class=\"whitespace-normal break-words\">Speed can remain constant even while velocity changes, because velocity depends on both speed and direction<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-563","chapter","type-chapter","status-publish","hentry"],"part":424,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/563\/revisions"}],"predecessor-version":[{"id":1712,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/563\/revisions\/1712"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/424"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/563\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=563"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=563"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=563"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}